\section{Results}
\label{sec:result}

Question 1:

Ziwei's Email: As you said, "One important result is that 2-3 collisions are all needed to build up v2, and more collisions do not help building up further elliptic flow (maybe even reducing it)". This may be motivated by Fig.5; however I think more analysis is needed to see whether this is indeed the case, because the change of v2 for partons with Ncoll=3 from their collision $\#1$ may be different from that for partons with Ncoll=1. For example, for soft partons (<0.5GeV) with Ncoll=3, it could be that collision $\#1$ gives a big increase of v2 and then collision $\#2$ and/or $\#3$ decreases v2; this would support your above statement; but it also could be that each subsequent collision continues to increase the v2 (certainly we may expect that a later collision increases v2 less than an earlier collision).

Here is the plot for comparison. \ref{fig:v2_vs_N_1} is the v2 vs N function. Here the $P_{T}$ range comes from final partons and the N means the parton totally has N collsions. However in \ref{fig:v2_vs_N_2}, N means the parton which already has N collsions. 
As we can see from \ref{fig:v2_vs_N_1} the v2 does not change after about 3 collisions as we expected before.
\begin{figure}[!htbp]
  \begin{center}
  \subfigure[v2 vs totol N]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_1.pdf}
    \label{fig:v2_vs_N_1}
  }
  \subfigure[v2 vs N]{
    \includegraphics[width=0.45\columnwidth]{figures/v2_vs_N_2.pdf}
    \label{fig:v2_vs_N_2}
  }
  \end{center}
\end{figure}

\clearpage

Question 2: 

\ref{fig:ncoll} shows that hard partons have more collisions than soft partons; this was explained as "because they have more energy to scatter". I would argue otherwise. Since partons at all Pt have the same cross section in AMPT and there is no lower threshold where parton collisions are forbidden, having more energy for a parton does not lead to more collisions. Instead, I think this is the formation time effect, since hard partons are formed earlier and thus can start scattering earlier and have more collisions; note that we set a parton formation time from string-melting as $tf=E_H/m_{T,H}^2$ where H represents the parent hadron of the parton.

You are right. In \ref{fig:pttime}, we confirm the relationship between parton pt and formation time. It's as we expected.

\begin{figure}[!htbp]
  \begin{center}
  \subfigure[Noll]{
    \includegraphics[width=0.9\columnwidth]{figures/Ncoll.pdf}
%    \caption{The differencial distribution of number of collision in different $P_{T}$ region.}
    \label{fig:ncoll}
  }
  \subfigure[pTtime]{
    \includegraphics[width=0.9\columnwidth]{figures/pt_time.pdf}
    \label{fig:pttime}
  }
  \end{center}
\end{figure}


\clearpage
Question 3:

For \ref{fig:dpt_vs_N} it's stated that "The trend is always partons gain less or loss more energy as the number of collision is larger". For high Pt partons, I agree that the trend is that they loss more energy as the number of collision is larger. However, for low Pt partons, I think the trend is that they gain MORE energy as the number of collision is larger. These trends are just expectations due to thermalization, and my "low" and "high" Pt scales here are defined relative to the mean Pt of all partons <$P_{T}$>~0.4GeV. I think Fig.3(a) shows a different trend because those partons (initial Pt <0.5GeV) contain both low and high Pt components, so the two expected trends mix together and produce something different.

To verify this, one can plot a similar figure as Fig.3 but for really-low-Pt partons (say initial Pt <0.2 GeV), and I suspect that their ($P_{T}^{final}-P_{T}^{initial}$) will increase and then gradually flatten out vs Number of Collision; just like Fig.3(b,c,d) where the curves for high Pt partons decrease and then gradually flatten out vs Number of Collision.

If the above for really-low-Pt partons turns out true, one can do the following further analysis: plot the ratio
($P_{T}^{final}-P_{T}^{initial})/(P_{T}^{initial}-<P_{T}>$)
for Pt<0.2 GeV, >1GeV, >2GeV and >3GeV.
This quantity is like the fractional energy loss/gain, but the good thing about using ($P_{T}^{initial}-<P_{T}>$) here is that it is negative for low Pt partons but positive for high Pt partons. On the other hand, since I expect ($P_{T}^{final}-P_{T}^{initial}$) to be positive for low Pt partons but negative for high Pt partons, this ratio will always be negative for any Pt. I suspect that the four curves are similar in shape, and it would be really fun if they are also similar in magnitude (then the curve would be ~universal).

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/dpt_vs_N.pdf}
  \caption{$P_{T}^{Final}-P_{T}^{Initial}$ as a function of number of collision in different $P_{T}$ region.}
  \label{fig:dpt_vs_N}
  \end{center}
\end{figure}

In \ref{fig:ptn02} is the plot as \ref{fig:dpt_vs_N} but the $P_{T}$ shreshold is 0.2GeV. 

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/ptn02.pdf}
  \caption{$P_{T}^{Final}-P_{T}^{Initial}$ as a function of number of collision in $P_{T}<0.2GeV$ region.}
  \label{fig:ptn02}
  \end{center}
\end{figure}

In \ref{fig:rpt_vs_N}, we show $P_{T}^{final}-P_{T}^{initial})/(P_{T}^{initial}-<P_{T}>$ as a function of N. The $<P_{T}>=0.308GeV$. In different $P_{T}$ region they have similar curves as we expected but not same magnitude. 

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/rpt_vs_N.pdf}
  \caption{$P_{T}^{final}-P_{T}^{initial})/(P_{T}^{initial}-<P_{T}>$ as a function of number of collision in different $P_{T}$ region.}
  \label{fig:rpt_vs_N}
  \end{center}
\end{figure}


\clearpage 
Question 4: 

\ref{fig:pt2d} shows that some initial low Pt partons gain energy to become high Pt. Although possible, I would expect a very small probability, so I'm surprised to see many final high Pt partons are generated this way; for example, for partons with final Pt between 6-7 GeV, I see ~5-10$\%$ coming from initial Pt below 1 GeV. So I'd like to confirm that this is not an artifact of tracking uncertainty when 2 identical partons interact, e.g. u(10GeV) u(2GeV) -> u(8GeV) u(4GeV) is counted as u(10GeV) -> u(8GeV) and  u(2GeV) -> u(4GeV), not as u(10GeV) -> u(4GeV) and u(2GeV) -> u(8GeV). Note that for each interaction in parton-collisionsHistory.dat parton $\#1$ is the parent of parton $\#3$. If the tracking analysis has no problem, then it may be another important finding that so many high Pt partons can come from elastic scatterings of low Pt partons.

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/pt2d.pdf}
  \caption{2D plots for initial and final parton $P_{T}$.}
  \label{fig:pt2d}
  \end{center}
\end{figure}


In order to check this problem, we use the sample that the parton's initial $P_{T}<1GeV$ and final $P_{T}>3GeV$. 
In \ref{fig:ptn} is the pt distribtuion vs number of collisions in this sample. Here N means how many times parton has collided not total number.
Another strange thing is that the parton always gain much energy at one collsion. We can see it from \ref{fig:maxdpt}, it shows the max momentum the parton gains in all collisions. There is a peak ( I thought it should be sharper) at about 3GeV which shows alot of partons' energy change from 1GeV to 3GeV by just 1 collision.
\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/checktracing_ptn_sct.pdf}
  \caption{2D plot of parton $P_{T}$ at different N collisions.}
  \label{fig:ptn}
  \end{center}
\end{figure}

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/dpt_max.pdf}
  \caption{Max $\Delta P_{T}$ of parton.}
  \label{fig:maxdpt}
  \end{center}
\end{figure}
  
In the \ref{fig:case} is an example of the parton energy gaining. 

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/case1.jpg}
  \label{fig:case}
  \end{center}
\end{figure}

A further check about the $P_{T}$ transportation is \ref{fig:max}. It's the fraction of max  $\Delta P_{T}$ to total $\Delta P_{T}$. The sample is partons with more than 1 collision. And the red line is the partons with $P_{T}^{initial}>3GeV$ the black line is partons with $P_{T}^{initial}<0.5GeV$. The parton which has only one time very large energy gain should be at $\Delta P_{T}^{max}/\Delta P_{T}=1$. But as we can see from \ref{fig:max}, the most of partons distribute from 0.5 to 1 that means two things. The first thing is most of the partons have more than one time "much-energy-gaining" collision. The second thing is that even low initial partons have energy loss for the $\Delta P_{T}^{max}/\Delta P_{T}>1$ region of the black line.

\begin{figure}[!htbp]
  \begin{center}
  \includegraphics[width=0.9\columnwidth]{figures/maxpt_dpt.pdf}
  \label{fig:max}
  \end{center}
\end{figure}
